Question

a small business is selling custom mugs. the business owner has observed that if they charge $5 or $25 per mug, they make no profit. the table below shows additional expected profits based on various prices for the mug. create a quadratic function of the form $p(x) = a(x - m)(x - n)$, where $a$, $m$, and $n$ are integers, that models the business’s profit $p(x)$ in terms of the price $x$ per mug.

Explanation:

Step1: Identify roots (m and n)

The business makes no profit (P(x) = 0) at x = 5 and x = 25. So, m = 5 and n = 25. Thus, the quadratic function is \( P(x) = a(x - 5)(x - 25) \).

Step2: Find the value of 'a'

Use a point from the table, e.g., (15, 100). Substitute x = 15 and P(x) = 100 into the function:
\[
100 = a(15 - 5)(15 - 25)
\]
Simplify the right side:
\[
100 = a(10)(-10)
\]
\[
100 = -100a
\]
Solve for 'a':
\[
a = \frac{100}{-100} = -1
\]

Step3: Write the final function

Substitute a = -1, m = 5, and n = 25 into the form:
\[
P(x) = -1(x - 5)(x - 25)
\]
Or simplified:
\[
P(x) = -(x - 5)(x - 25)
\]

Answer:

\( P(x) = - (x - 5)(x - 25) \) (or expanded as \( P(x) = -x^2 + 30x - 125 \), but the factored form is requested)